Let
C be an ∞-category. If f, g : x \to y are morphisms (1-simplexes) in
C, then we write f \sim g if there is a 2-simplex \sigma : \Delta^2 \to C such that \sigma(0 \to 1) = f, \, \sigma(0 \to 2) = g, \, \sigma(1 \to 2) = \operatorname{id}_y. Then by Joyal's work, the relation \sim turns out to be an equivalence relation. Hence, we can take the quotient :[x, y] = \operatorname{Hom}_C(x, y)/\sim. Then the homotopy category \tau(C) in the sense of Boardman–Vogt is the category where \operatorname{obj}(\tau(C)) = \operatorname{obj}(C), \operatorname{Hom}_{\tau(C)}(x, y) = [x, y] and the composition is given by [f] \circ [g] = [h] when h exhibits some composition of f, g. Let \pi_0 be a left adjoint to the inclusion of the
category of sets into the category of simplicial sets. If K is a Kan complex, then \pi_0 K coincides with the set of
simplicial homotopy classes of maps \Delta^0 \to K. Then :\operatorname{Hom}_{\tau(C)}(x, y) \simeq \pi_0 \operatorname{Map}(x, y) for each objects x, y in C. == See also ==